Littré’s definition adds some more details, but is still unclear:

du latin tostus,
brûlé, par assimilation de la rapidité de la flamme ; c’est ainsi que,
dans l’ancien français, on a dit chaut pas, pour rapidement ; que
Dante a dit impresa tosta, entreprise rapide(…)

So Tostus means burnt/toasted, hence cooked fast and strong. Something done swiftly gets finished early, so I guess this how we get to the final meaning.

It seems not perfectly consistent with the example : if "chaut pas" (hot step) means walking fast, then "chaut/chaud" means fast and tostus must be "very fast". And there’s no obvious link between the rapidité de la flamme and "chaut pas".

The shift between fast and early is not totally satisfying either, but I can imagine a sentence like this:

Cette tâche doit être terminée aussi vite que possible.

This task shall be finished as quickly as possible.

shifting over time to mean:

Cette tâche doit être terminée aussi tôt que possible.

That task shall be finished as early as possible.

I’m not a linguist, though.